Is there any the law or procedure regarding conflicts of interest during an impeachment trial in the US Senate?
I can think of many for senators on both sides; obviously political conflicts, (favoring a policy over another, etc) and I understand the logic of overlooking those.

However, some examples, such as Mitch McConnell having a member of his family in the administration's cabinet seems different as he would indirectly lose financial income and/or influence. If the President is removed, his cabinet likely goes with him.

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Is there any the law or procedure regarding non-political conflicts of interest during an impeachment trial in the US Senate?

There is no law requiring, nor any procedure, for removal of a Senator for any conflict of interest. There is a procedure whereby a Senator could be excused at their own request. The Senator states a reason to be excused, then the Senate by a simple majority decides whether to grant the request.

Senators from time to time have asked to be excused from participation in an impeachment trial. During the trial of Halsted L.
Ritter, the Senator from Colorado (Mr. Costigan) asked unanimous
consent to stand aside from participation in the trial, with a statement of his reasons therefor entered in the Record, it was granted.

[...]

A total of twenty-one requests by Senators to be excused from voting were granted during votes on the articles involving Judge Louderback.1

Also worth mentioning that if there was some kind of vote on stopping other senators from voting on the matter of conviction, that would also be subject to conflicts of interest. Likely more than the conviction vote in the first place.
– bobsburnerJan 24 at 13:23